The illusion of control is not limited to the placebo effect. When you consider that our Universe is causal and that all actions (including that of a human) result from a set of conditions determined by previous actions, it is quite easy to discount the existence of free will.

Without getting existential, I would like to raise that not only does pushing a placebo button give you no control, but the choice even to press that button was not one you took but one which was predefined. This means that even if that button wasn't a placebo, you would still ultimately have no control over what happens as the choice to push the button is not yours.

This extends further questions as to who you are as you are defined by your actions which were not your choice to make. This obviously has serious implications with our legal system where by a person could not be held accountable for commiting a crime however it is also worth noting that the notion of awards would be invalidated as a person had no choice in doing good thing X.